Improvement Of Precise Integration Method And Its Application In Dynamics And Control

Numerical computation for a set of simultaneous ordinary differential equations(ODEs) is very important in applications. So far, tremendous efforts have been devoted to finding appropriate numerical methods to solve this problem. In recent years, precise integration method(PIM) for the numerical integration of ODEs has been proposed and has attracted a wide range of concerns. PIM not only can give a high accurate numerical result, which approaches the computer precision for the linear time-invariant ODEs, but also is free from stiff problems. It has been extended to solve time-variant, non-linear and partial differential equations and successfully applied to various fields, such as structural dynamics, random vibration, wave prorogation, transient heat conduction and optimal control, et al. The PIM provides a basic algorithm platform with high precision and high stability, so it is worthy of studying further. At the same time, numerical awareness in control needs to be increased. Especially, it is of great importance to choose an appropriate theoretical framework for constructing algorithms of high performance. The starting point of the state space method, i.e. the basis of modern control theory, should trace back at least to the Hamiltonian canonical equation system, which demonstrates that structural mechanics and optimal control have the same mathematical basis. Along this way of consideration, the mathematical problems of the two different fields have a one-to-one correspondence with each other. As a result, it is helpful to make further researches on the controller design and its numerical computation in optimal control field by introducing the mature methods in mechanics, such as finite element method, sub-structure techniques, et al. This dissertation aims at the development of efficient and reliable numerical methods, improves the performance of the PIM-based algorithm platform and studies the problems of the controller design and its numerical computation arising from the time-delay, time-varying, and non-linear optimal control systems. This dissertation also develops an optimal control system design and simulation toolbox (PEMCSD Toolbox) and applies it to the study of the formation flying control system. The main research work covers the following aspects:(1) This dissertation presents an iterative algorithm for adaptive selection of the scaling parameter N and series-expanding parameter q by using the approximation theory of matrix functions, which plays an important role on the computational efficiency and numerical accuracy of the PIM based on Padéapproximation for matrix exponential. The proposed algorithm not only improves the computational efficiency, but also is independent of the matrix's characteristics. Numerical tests are made by comparing with the MATLAB's built-in function expm(), and it shows that the proposed algorithm achieves the same efficiency, but higher accuracy and stability.(2) This dissertation presents the extended precise integration method(EPIM) for computing the response matrices of Duhamel integrations arising from non-homogenous dynamic systems. Numerical results of the response matrices can approach the computer precision when the non-homogenous terms are of the following forms, such as polynomial, exponential, trigonometric functions and their combinations. More importantly, the EPIM is independent of the quality of the system matrix(or its relative matrices) since it dose not need the inverse matrix calculation. The EPIM has been applied to some problems: 1) Combined with the pseudo-excitation method, an efficient and accurate algorithm for computing random responses of structural vibration has been proposed; 2) Combined with the traditional numerical integration techniques, such as the single-step method based on Taylor series expansion and the Adams multi-step method, several numerical algorithms with high efficiency and accuracy for the solution of non-linear differential equations have been constructed; 3)Making full use of the characteristics of periodicity, researches on computing the Floquet transition matrix of the periodic time-varying system and solving the responses of a class of periodic non-linear system have been made. Numerical examples show that the EPIM-based algorithms have advantages of high accuracy, high stability and simple formulas, which greatly improve the numerical stability and expand the scope of applications.(3) The algebraic-equation method and interval-elimination method are derived based on the interval analysis techniques and the extended precise integration method(EPIM) for computing the interval response matrices, arising from the non-homogenous terms of the two point boundary value problems(TPBVPs), is proposed. The EPIM can give high precise numerical results approaching to computer precision for non-homogenous terms with certain special forms. High accurate and efficient algorithms based on the EPIM are constructed for the problems with general non-homogenous terms, infinite interval and time-varying coefficients. Finally structure-preserving algorithms for certain matrix differential equations with periodic coefficients, such as Riccati equations, Lyapunov equations and Sylvester equations, are constructed by combining interval analysis method with the EPIM for periodic Floquet transition matrix. Numerical results verify the effectiveness of the algorithms.(4) Robust H_∞optimal control and filter of time-delay systems are studied in a uniform frame work. Firstly the continuous time-delay systems and its performance indices are discretized by the EPIM in order to ensure the equivalence to its original systems as much as possible. Then the discrete time-delay systems are transformed into standard discrete forms without time-delay by introducing the appropriate extended state vectors. So theories and methods of usual discrete system can be applied. Taking into account the increasing dimensions of the standardized system, the interval mixed energy method and extended W-W algorithms are introduced, which are with high parallelism and stability. So a set of accurate and stable algorithms are proposed for the computational problems of H_∞optimal control and filter systems. A H_∞full information controller with control time-delay is designed and applied to vibration attenuation of the seismic-excited buildings. Simulation results show that structural vibration is greatly attenuated for different amounts of time-delays and different types of seismic excitation, which verifies the effectiveness of the controller.(5)This dissertation presents the symplectic-conservative perturbation method for computational problems of linear time-varying and non-linear optimal control systems. Since the necessary conditions of optimal control problems are equivalent to the TPBVPs of Hamiltonian systems, its numerical methods should be symplectic conservation. Firstly the symplectic conservative perturbation method based on the PIM is presented for the linear time-varying TPBVPs. Combination formulas between the zeros-order system and the perturbation system are derived based on the interval mixed energy matrices and the transition matrices, respectively. Their relationships are investigated and the former is found to be a better choice for optimal control problems because of its inherent stability. The symplectic-conservative perturbation method for non-homogenous time-varying Hamiltonian TPBVPs is further proposed and applied to the iterative computation of the non-linear optimal control problems. Numerical results show it not only increases convergency of the iteration algorithm but also decreases sensitivity to the initial iterative values greatly, which demonstrates that the proposed method is both high accurate and symplectic-conservative.(6) Due to high feedback gains or singularity at the terminal time of traditional terminal controllers, an open-loop control for a short interval before the end time is often adopted. This dissertation presents a non-singular terminal controller with the feedback-feedforward architecture in both intervals, by introducing a new terminal "soft constraint" term to improve the variational formulas and using the essence of constant for the Lagrange multiplier. Influences of the "soft constraint" term on constructing the feedback controller are studied, which is of special importance for the minimal energy control. Closed-form solutions to the feedback matrices and states of the controlled system are constructed by introducing interval mixed energy matrices. Further more, structure-preserving algorithms are derived, which greatly facilitates the design and implementation of terminal controllers. The proposed method has been extended to the terminal controller of discrete-time systems successfully.(7) The optimal control system design and simulation toolbox(PIMCSD Toolbox) is developed in view of the absence of functions on the finite time optimal control. Then time-varying controllers for the typical double satellites formation reconfiguration are studied based on the PIMCSD Toolbox. Research results provide an important reference for engineering designs and applications of spacecraft formation control systems.